An equation holds for its solutions; an identity holds for every value in its stated domain. The symbol can emphasise an identity. A proof links known facts to the required conclusion; numerical tests alone do not prove a universal claim. **Task** | **Start** | **Check** Prove an identity | Simplify one side until it becomes the other. | Keep every denominator nonzero. Prove a numerical fact | Use exact arithmetic. | Avoid rounded approximations in an exact proof. Check a result | Substitute into the original expression or reverse a valid calculation. | A successful check is evidence for that case, not a general proof. Example For x!= -1, prove x2-1x+1 x-1. LHS= (x-1)(x+1)x+1 =x-1 =RHS. The restriction x!= -1 is essential: cancelling a factor does not make the original fraction defined at -1.

An equation holds for its solutions; an identity holds for every value in its stated domain. The symbol \(\equiv\) can emphasise an identity. A proof links known facts to the required conclusion; numerical tests alone do not prove a universal claim.

Simplify one side until it becomes the other.

Avoid rounded approximations in an exact proof.

Substitute into the original expression or reverse a valid calculation.

A successful check is evidence for that case, not a general proof.

For \(x\ne-1\), prove \(\dfrac{x^2-1}{x+1}\equiv x-1\).

\(\begin{aligned}\text{LHS}&=\dfrac{(x-1)(x+1)}{x+1}\\&=x-1\\&=\text{RHS}.\end{aligned}\)

The restriction \(x\ne-1\) is essential: cancelling a factor does not make the original fraction defined at \(-1\).

Mathematical Induction

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Deductive proof and identities

Deductive proof and identities

Task	Start	Check
Prove an identity	Simplify one side until it becomes the other.	Keep every denominator nonzero.
Prove a numerical fact	Use exact arithmetic.	Avoid rounded approximations in an exact proof.
Check a result	Substitute into the original expression or reverse a valid calculation.	A successful check is evidence for that case, not a general proof.